Action operads and the free G-monoidal category on n invertible objects

We use the theory of action operads and their algebras to study a class of associated monoidal categories, particularly those that are freely generated by some number of invertible objects. We first provide some results about G_n, which is known to be both the free EG-algebra and the free G-monoidal category over n objects, for a given action operad G. Then we deduce the existence of LG_n, the free algebra on n invertible objects, and show that its objects and connected components arise as a group completion of the data of G_n. In order to determine the rest of LG_n, we will prove that this algebra is the target of a surjective coequaliser q of monoidal categories; that collapsing the tensor product and composition into a single operation forms one half of an adjunction M(_)^ab ⊣ B; that its action operad G embeds into its group completion; and that its morphisms are a semidirect product (s × t)(LG_n) ⋉ LG_n(I,I) of a chosen subgroup by the unit endomorphisms. With these and other assorted results, we will compile a method for constructing LG_n for most action operads, and from this produce descriptions of the free symmetric, braided, and ribbon braided monoidal categories on invertible objects.